How is the sum of a rational and irrational number irrational?

Answer 1

This can easily be proved by contradiction. Without loss of generality, I will take specific numbers as an example. The proof can easily be extended to any rational + irrational number.

Assumption: 1 plus the square root of 2 is rational. (It is a well-known fact that the square root of 2 is irrational. No need to prove it here; you can use any other irrational number will do.)

This rational sum can be written as p / q, where "p" and "q" are whole numbers (this is basically the definition of a "rational number").

Then, the square root of 2, which is equal to the sum minus 1, is:

p / q - 1

= p / q - q / q

= (p - q) / q

Since the difference of two whole numbers is a whole number, this makes the square root of 2 rational, which doesn't make sense.